Using first-principles calculations, we investigate the structural stability and electronic properties of the smallest (2,2) carbon nanotube as the inner wall of the double-walled carbon nanotubes. The results indicate that an isolated (2,2) carbon nanotube is unstable upon unrolling in free space, while it can exist within (6,6) and, especially, within (7,7) carbon nanotubes. In addition, a spontaneous symmetry breaking of the isolated (2,2) nanotube produces an energy gap at Fermi level converting it into a semiconductor.